Many radio frequency sensor systems with digitally implemented signal processing functions require analog-to-digital converters (ADCs) for converting received bandpass signals into digital form. The quality of the performance of the digital system often depends upon the analog-to-digital converter's dynamic range, sampling jitter, and conversion rate, as a function of input signal frequency, where the output converted signal is sampled at a rate that is at least twice the bandwidth of the signal, according to the Nyquist criterion. The dynamic range is defined by the signal to noise and distortion ratio and the spurious free dynamic range, where the noise and distortion terms produced by the ADC corrupt the received signals and limit the ability to discern small signals from larger interference signals in the correlative and interference suppressing digital signal processing functions typically performed down-stream of the digital receiver. Examples include channelization filtering, digital beam forming, pulse Doppler filtering, and moving target detection. The processed residues are composed of receiver noise and spurious terms that are often dominated by that produced within the ADC element. An ideal digital receiver produces processed residues composed solely of thermal noise associated with the antenna characteristics and background radiation.
Some types of analog-to-digital converters employ multiple parallel decision circuits to resolve finely spaced levels. These circuits require well-matched input stages to achieve satisfactory signal to noise and distortion ratio. A typical approach for sampling and converting signals several hundreds of megahertz in bandwidth, or greater, employs this class of technique implemented with high speed mixed signal integrated circuit technology, where typically the transistors are capable of current gain at frequencies greater than 20 GHz, and can yield monolithic circuits with transistor count between 1000 and 20000. In this class of ADC, a feedforward multistage architecture is used where the first stage resolves the most significant bits and a second stage resolves a sampled analog difference signal produced by subtracting the digital-to-analog converted result of the first stage from a delayed sampled analog input. It is difficult to attain low noise decision circuits and sampling circuits with precise distribution and matching, and sampling aperture uncertainty to enable the desired signal to noise ratio and spurious free dynamic range levels. Increasing resolution in this type of ADC requires an exponentially increasing number of decision circuits and exponentially more stringent component and signal distribution matching requirements, that include small amplitude and delay skews. In general, to resolve N bits of amplitude of a given sample the aggregate linearity, noise, jitter and dynamic uncertainties must combine to a total error term that is (2Nbit−1)/√12. Hence, resolving to a 16-b level for example, will require all errors to combine to 4.4 parts per million of the maximum output level minus the minimum output level.
Sigma-delta analog-to-digital converters achieve high resolution and low noise because the underlying technique utilizes a feedback loop to suppress the noise of a low resolution quantizer over a given pass band. Typically, suppression between 20 dB and 90 dB can be obtained with high order loop filters and with weights that spread the noise shaping zeroes over the pass band. As a result, a much smaller number of parallel decision circuits are required to attain high resolution, with relaxed matching requirements; however, these circuits must be capable of much greater bandwidth than that of the signal. To achieve high signal-to-noise ratio and large spurious dynamic range, the feedback DAC must be designed for low noise and jitter, and the input sum node and loop filter must be sufficiently linear and low in input equivalent noise to enable the improved dynamic range sought. The DAC nonlinearity can be corrected digitally, when multiple bits are employed, by means of Nonlinear Filter Error Correction as described in U.S. Pat. No. 6,271,781, issued in the name of Pellon. To achieve a particular resolution, full precision is not required in the DAC as a result of oversampling and uniquely adapted error correction for each DAC; however, the highest DAC precision is desirable since the degree of nonlinearity error correction required is reduced, and thus, the error that would result from changes in the DAC nonlinearity produced through temperature changes in the operating environment. Other techniques have also been developed, known as Dynamic Element Matching, which also be used to mitigate multibit DAC error effects.
Finally, to minimize latency, the DAC is implemented within a highly integrated ADC-DAC that must also exhibit sufficiently low noise densities associated with the metastable or ambiguous response of the ADC to signal inputs near its thresholds. The latency required to achieve the required DAC spectral purity can often exceed a clock cycle.
The improvement in quantization noise dynamic range due to loop filtering reduces as the pass band bandwidth is increased, where the noise shaping zeros, produced by the feedback loop, must be spread over a greater pass band. This spreading reduces the mean attenuation attained. Increasing the number of zeros can improve performance but with diminishing advantage as the loop filter order is increased. Stability is affected, particularly for pass bands that are a significant fraction of the Nyquist frequency, such as ⅛ to ¼. In prior art continuous-time sigma-delta modulators, the bandwidth is set relative to a frequency which is the inverse of the loop latency. Ideal continuous-time sigma-delta modulation requires a single whole clock cycle delay equal to one sampling interval; however, in practice, the actual delay available is based on the sum of component latencies, which can exceed several clock cycles, and is not a whole number. This is particularly true when, for a given transistor technology, the highest possible sampling rate in the quantizer and the highest possible amplifier bandwidth in the loop filter are employed to achieve improved pass band bandwidth. Prior art techniques exist for compensating the effect of this latency to achieve stable operation with control over the pole and zero locations; however, the noise shaping attenuation as a function of pass band bandwidth is reduced as a result of this latency.
One such sigma-delta configuration includes a degenerative feedback loop including an analog-to-digital converter which digitizes the feedback signal and a digital-to-analog converter which generates an analog version of the digital output signal, together with a differencing circuit which subtracts the analog version of the digital output signal from the input signal to form a difference or error signal. This error signal is resonated by a recursive transversal implementation of the loop filter and applied to the underlying low resolution sampling analog-to-digital/digital-to-analog converter (ADC-DAC). The degeneration occasioned by the negative feedback tends to suppress the inherent noise of the underlying quantizing ADC, as well as tending to correct the value of the digital signal toward that of the analog input signal. This configuration employs transversal delays set in accordance with the latency achieved in the primary feedback loop. The latency of the sum node, loop filter and the quantizing ADC-DAC are combined to form the minimum realizable transversal delay. The inverse of this delay is termed the maximum periodic frequency of the recursive transversal filter. The attenuation as a function of bandwidth is determined as a fraction of this periodic frequency. This is often much less than the sampling frequency as a result of the component latencies, when designed for sufficient linearity, isolation and spectral purity. This is particularly true when considering very high clock rates, such as four or twenty gigasamples per second, higher order multistage loop filters that typically have orders between 2 to 10 and multibit quantizing ADC-DACs with between 2 to 8 bits of resolution.
FIG. 1 is a simplified block diagram of a sigma-delta (ΣΔ) analog-to-digital converter or encoder as described in U.S. Pat. No. 5,608,400, issued Mar. 4, 1997 in the name of Pellon. IN FIG. 1, analog input signals x(t) which are to be converted to digital form are applied to an input port 12. The analog input signals are band-limited, as would be the case, for example, if they were modulated onto a radio-frequency (RF) or intermediate-frequency (IF) carrier. FIG. 2a illustrates an idealized amplitude—(A) versus-frequency plot of a band-limited analog input signal spectrum 210 centered on an IF frequency. The analog input signals applied to input port 12 are coupled to a first input port 14i1 of a summing circuit 14, where they are combined with a signal applied to a second input port 14i2 of summing circuit 14 by way of a signal path 16. The summed signals are coupled from summing circuit 14 by way of a path 18 to a first input port 20i1 of a further summing circuit 20. Summing circuit 20 sums the signals applied to its first input port 20i from signal path 18 with analog signals applied to its second input port 20i2 from a signal path 22, to produce a summed signal at its output port 20o, which is coupled onto signal path 24. Summing circuits 14 and 20 together may be viewed as a three-input-port summing circuit, which produces a summed signal on signal path 24. The three input ports, taking this view, are 14i1, 14i2, and 20i2.
A coupling arrangement 26 of FIG. 1 receives signals applied from signal path 24 at its input port 26i, and couples them to an output port 26o and a signal path 28. Coupling arrangement 26 also samples the signals traversing from its input port 26i to output port 26o to a tap or sample port 30, for reasons described below. The signals coupled from output port 26o of coupling arrangement 26 onto signal path 28 are applied to an input port 32i of an analog-to-digital converter (ADC) 32, which is sampled at a clock rate by a clock signal applied to a clock input port 32c from a signal path 34. Analog-to-digital converter 32 samples digital signals applied to its input port 38i at the clock rate, and converts the signals into digital form. Those skilled in the art know that the noise of ADC 32 may be represented as a noise signal e(n) applied to a effective noise input port 32n, and that such an ADC should have low latency or time delay, and that it may be desirable to sacrifice resolution (number of bits) to achieve higher speed. The digital signal is generated as yn on output port 32o. The digital signal from port 32o is applied to a system output port 40o, and is also applied by an output signal path 40 to an input port 38i of DAC 38. A clock signal generator 36 generates clock signals at a clock frequency fs, which are applied by way of signal path 34 to clock input port 32c of ADC 32 and to a clock input port 38c of a digital-to-analog converter (DAC) 38. Those skilled in the art know that the noise of the DAC 38 may be represented as a noise signal g(n) applied to a effective noise input port 38n. 
Digital-to-analog converter 38 of FIG. 1 receives the digital signal applied over signal path 40 from output port 32o of ADC 32, and converts the digital signal into analog form at the clock rate. As is known to those skilled in the art, a zero-order hold spectral response, illustrated as a block 42, is generally inherent in DAC 38. DAC 38 produces its analog signal output at an output port 38o. An amplitude equalizer 44 is provided. Equalizer 44 has an amplitude response selected to flatten or compensate, up to a cutoff frequency fCO, for the magnitude response of the zero-order hold 42 of DAC 38, as described in more detail below. Equalizer 44 is coupled to the second input port 14i2 of summing circuit 12 and to output port 38o of DAC 38 by a coupling circuit, which includes at least one direct-current (DC) blocking or AC-coupling path. As illustrated in FIG. 1, the coupling circuit includes a capacitor 46 coupled to output port 38o of DAC 38 and to input port 44i of equalizer 44, together with an all-pass conductive signal path 16 coupled between output port 440 of equalizer 44 and second input port 14i2 of summing circuit 12. The positions of capacitor 46 and all-pass conductor 16 could be interchanged, if desired. The described connections establish a main degenerative loop suggested by an arrow 48. In a preferred embodiment of the structure of U.S. Pat. No. 5,608,400, the equalizer is an elliptical filter.
The sigma-delta analog-to-digital converter 10 of FIG. 1 also enhances the open-loop gain of main loop 48 at a frequency, which as described below is selected to include the IF frequency (or any other desired frequency) at which the analog input signal is centered. The open-loop gain of the main loop 48 is enhanced by a regenerative or positive-feedback loop (resonator), suggested by an arrow 50, embedded in the main loop. Regenerative feedback loop 50, as illustrated in FIG. 1, includes a low-pass filter (LPF) 52, an aliasing resonance null filter 54, and a DC blocking or AC-passing capacitor 56, coupled in cascade, by conductors designated generally as 22, between tap port 30 of coupling arrangement 26 and input port 20i2 of summing circuit 20. The cascade of LPF 52, null filter 54, and DC blocking capacitor 56, is interconnected by a delay line 58, conductors 22, and a loop gain element or amplifier 60. The regenerative loop 50 thus includes not only delay line 58, LPF 52, null filter 54, amplifier 60, and capacitor 56, but also includes a portion of summing circuit 20, together with that portion of coupling arrangement 26 which lies between input port 26i and tap port 30. The portion of coupling circuit 26 included within regenerative loop 50 contains a phase (φ) shifter 62. Coupling circuit 26 also includes a gain element 64 associated with main degenerative feedback loop 48 and a further delay element 66.
Low-pass filter 52 of FIG. 1 has an amplitude response which is selected to be equal to the open-loop response of main degenerative feedback loop 48, up to a cut-off frequency fCO. FIG. 2b illustrates a simplified response 222 of low-pass filter 52, with cut-off frequency of fCO, and with the effect of DC blocking capacitor 56 near zero frequency. In general, the response of LPF 52 includes a flat or constant amplitude portion up to the cut-off frequency (frequency at which the amplitude response begins to decrease), which lies somewhat above the IF frequency of FIG. 2a. The Nyquist frequency (one-half of the clock frequency) is illustrated as fN.
Closing of the regenerative loop 50 of FIG. 1 will generate a gain pole at a selected resonance frequency, which is one of a comb of resonance frequencies. The period of the comb, which is the frequency interval between “teeth” of the comb, is the inverse of the delay around the regenerative loop 50, and thus is affected by the delay inherent in the components of the loop, the interconnection conductors of the elements of the loop, and by the time delay of delay element 58. FIG. 2c illustrates the open-loop amplitude-frequency resonant peak response of the regenerative feedback loop 50, in the absence of filters 52 and 54; the response defines a comb response 212 including a selected or first response peak 214 at the IF frequency at which the analog input signal is centered, and two additional response peaks 216 and 218 of the comb. Response peak 218 is illustrated as being centered on a frequency fA. Near zero frequency, response 212 of FIG. 2c decreases to zero amplitude due to the presence of AC coupling capacitor 56. The response in the absence of AC coupling is illustrated by dash line 220. The attenuation of the closed-loop bandwidth above frequency fCO by the characteristics of filter 52, illustrated in FIG. 2b, enhances stability by reducing the importance of the tolerances of components of the continuous-time filters.
The response of aliasing resonance null filter 54 of FIG. 1 is illustrated as 226 in FIG. 2d. As illustrated therein, plot 226 exhibits a null at frequency fA, the same frequency at which the third resonant peak 218 of the regenerative response 212 occurs.
FIG. 10 illustrates a phase versus frequency plot 1010, which represents the step phase shift produced by phase shifter 62 of regenerative loop 50 of FIG. 1. The phase shift φ illustrated by plot 1010 has a constant value above zero Hertz, and a corresponding negative value below zero Hertz. This response is ideally
                                              ⁢                                                                                                                        H                      ϕ                                        ⁡                                          (                                              j                        ⁢                                                                                                  ⁢                        ω                                            )                                                        ≈                                    ⁢                                                            1                      2                                        [                                                                                            (                                                      1                            +                                                          sign                              ⁡                                                              (                                ω                                )                                                                                                              )                                                ⁢                                                  ⅇ                                                                                    -                              j                                                        ⁢                                                                                                                  ⁢                            ϕ                                                                                              +                                                                                                                                                              ⁢                                                            (                                              1                        -                                                  sign                          ⁡                                                      (                            ω                            )                                                                                              )                                        ⁢                                          ⅇ                                              j                        ⁢                                                                                                  ⁢                        ϕ                                                                              ]                                                                                                      =                                    ⁢                                                            cos                      ⁡                                              (                        ϕ                        )                                                              -                                          j                      ⁢                                                                                          ⁢                                              sign                        ⁡                                                  (                          ω                          )                                                                    ⁢                                              sin                        ⁡                                                  (                          ϕ                          )                                                                                                                                                                                                            (        1        )                            where        
          ⁢            sign      ⁡              (        ω        )              =          {                                                                  +                1                            ,                              ω                >                0                                                                                        0              ,                              ω                >                0                                                                                                        -                1                            ,                              ω                <                0                                                        Also illustrated in FIG. 10 is the phase shift 1012 introduced by delay element 58 of FIG. 1, which, as known, slopes with frequency. Plot 1014 of FIG. 10 illustrates the net phase response around the regenerative loop. If the magnitude of the step phase shift is increased, the offset of the net phase shift increases at all frequencies, thereby affecting the frequency at which the resonant loop resonates.
The open-loop gain of the regenerative loop 50 of FIG. 1 is set to unity, and the open-loop phase is set to zero, or to an integer multiple of 360°, at the selected resonance frequency, which in FIG. 2c is the frequency of the first resonance peak 214. However, the selected resonance frequency could be selected to be the frequency of one of the other resonance peaks of the comb, such as peak 216 or peak 218. Setting of the open-loop gain is accomplished by adjusting the gain of loop gain amplifier 60 of FIG. 1, or alternatively by applying a selected amount of attenuation in the regenerative loop. The open-loop phase is set to 0° or to N times 360° by adjusting the combination of the value of delay line 58 and/or the phase of phase shifter 62. The value or magnitude of the delay, without the phase-shifter, establishes the base frequency of the first or lowest-frequency resonance (the frequency of resonance 214 of FIG. 2c), and the frequency increment between each resonance and the next higher resonance (the comb interval). The addition of a phase-shift over and above the delay moves the resonant frequency of the first or lowest-frequency resonance, without changing the interval between each resonance and the next. For purposes of the description of U.S. Pat. No. 5,608,400, the term “phase-shift” or “phase-shifter” includes elements or circuits which accept a real-valued signal at their inputs, and which produce real-valued signals at their outputs, with a constant phase value (or constant value of phase shift) therebetween as a function of frequency at frequencies greater than zero Hz. At negative frequencies, these phase shifters mirror the phase response at positive frequencies, but with a negative sign. Such phase shifters may be implemented by all-pass ladders or lattice filters, as known in the art, but the exact method of implementation is not critical.
It should be noted that phase shifter 62 of FIG. 1 is located in both the main and the regenerative loops, but could alternatively be located solely in the regenerative loop, to give individual control to the regenerative loop phase, so long as another phase shifter were to be located solely in the main degenerative loop. In special cases, and especially in higher-order resonators, described below, phase shifter 62 is dispensed with entirely, and the desired phase shift is established by the cumulative phase shifts of the other elements of the main and regenerative loops. This is because the phase shifter circuits introduce phase nonlinearity and magnitude error in the form and phase and magnitude ripple error near the center frequency. Each particular higher order arrangement will have a corresponding set of gain and phase error margins, or margins relative to the stable condition of the closed-loop system. Elimination of the phase shifter elements in some of the arrangements assists in meeting the desired gain and phase margins, to thereby improve stability, and also has the advantage of a reduced parts count. In such arrangements lacking phase shifter elements, the gain is set as before, and the required phase of 0° or an integer multiple of 360° is obtained at the desired resonant frequency by the combined phase shifts of the components and adjusted delay element.
As a result of the sampling operation in ADC 32 of FIG. 1, components above frequency fN of FIG. 2c will alias to frequencies below fN. This aliasing will produce an effective resonance function other than that illustrated, representing that seen by the closed loop at the output of the ADC 32. The effective resonance will contain undesired gain components, which may affect the stability of the closed-loop sigma-delta ADC 10. Plot 228 of FIG. 2e illustrates an idealized amplitude-vs-frequency response 228 representing the result of combining the regenerative loop response 212 of FIG. 2c with the low-pass filter response 222 of FIG. 2b and the null-filter response 226 of FIG. 2d. The transfer function represented by plot 228 is
                              R          11                =                                            N              ⁡                              (                s                )                                                    D              ⁡                              (                s                )                                              =                                                    N                ⁡                                  (                  s                  )                                                            1                -                                  B                  ⁡                                      (                    s                    )                                                                        =                                                                                                      H                      ϕ                                        ⁡                                          (                      s                      )                                                        ⁢                                      e                                          -                      sTd2                                                                                        l                  -                                                                                    GH                        ϕ                                            ⁡                                              (                        s                        )                                                              ⁢                                                                  H                        B1                                            ⁡                                              (                        s                        )                                                              ⁢                                                                  H                        B2                                            ⁡                                              (                        s                        )                                                              ⁢                                          e                                              -                        sTd1                                                                                                        =                                                          (        2        )            where:                Hφ(s)=Laplace Transform of a Bandpass Phase Shifter Function;        HB1(s)=Laplace Transform of Filter Required to Match Primary Feedback Path;        HB2(s)=Laplace Transform of the Notch Filter For Suppressing the First Resonance above the Nyquist Frequency fN;        e−sTd132 Laplace Transform of Delay Line which combines with other delays in the regenerative loop to form the entire delay Te;        e−sTd2=Laplace Transform of Delay Line which combines with component delays in the primary loop to form the entire delay Te; and        B(s)=GHφ(s)HB1(s)HB2(s)=open loop response of the regenerative loops.As illustrated, the effect of the two filters on the regenerative loop response 228 is negligible at frequencies below cutoff frequency fCO. Above frequency fCO the cutoff of the low-pass filter progressively attenuates the response, and the resonance peak at fA is attenuated by the null filter response to a value below unity gain, illustrated as a level 230. This combination of responses tends to reduce all of the unwanted effective resonant peaks of the regenerative loop, restoring a stable condition. To suppress the effect of aliased components, the gain produced by the regenerative loop at frequencies greater than fN must be suppressed to a level sufficient for stability, which is represented by dot-dash line 230. In the process of transitioning from the desired high gain at the IF frequency to the low loop gain at fA, regenerative peak 216 can have a gain exceeding unity. Thus, plot 228 of FIG. 2e represents the closed-loop regenerative gain of the regenerative loop 50 of FIG. 1.        
In FIG. 2f, dot-dash plot 242 represents the open-loop transfer function seen by the closed-loop sigma-delta modulator, in the absence of a cutoff filter response, such as that of plot 222 of FIG. 2B. Plot 242 has an undesired peak at a frequency fz, where fz=2fN−fA, as a result of aliasing produced by the ADC and DAC, which sample at fS=2fN. This aliasing gain peak is undesired because it causes the gain and phase margins to be exceeded, thereby resulting in instability. In FIG. 2f, plot 244 represents the components produced by aliasing of components of plot 228 of FIG. 2e which lie above fN. Plot 244 has a response which is suppressed at all frequencies, attributable at least in part to the cutoff above frequency fCO, so that it has negligible effects on system performance. FIG. 2f represents as a plot 240 the effective open-loop response of the entire sigma-delta loop of the arrangement of FIG. 1. The term “effective” is used to indicate the open-loop transfer function seen by the closed-loop sigma-delta modulator, which differs from that of FIG. 2e as a result of the sampling process in the analog-to-digital converter 32 of the loop and the analog reconstruction process in the digital-to-analog converter 38. Plot 240 is the sum of components below fN of plot 228 of FIG. 2e with components represented by plot 244 of FIG. 2f. 
FIG. 3 illustrates the open-loop response of zero order hold 42 of FIG. 1 as a plot 310, the open-loop response of filter 44 as 312, and the combined response from input port 38i of DAC 38 of FIG. 1 to output port 44o of filter 44 as 314. Near zero frequency, the combined response 314 tends toward zero amplitude because of the effects of DC blocking capacitor 46. When both the regenerative loop 50 and the main degenerative loop 48 are closed, two transfer functions develop, a first signal transfer function Hx(ω), where ω is 2πf/fs, between the system signal input port 12 and the system output port 400, and the second noise transfer function, He(ω), between effective noise input port 32n of ADC 32 and system output port 400. Noise transfer function He(ω) will exhibit attenuation at frequencies at which open-loop resonance gain is high, as at frequency fIF of the selected resonance. Thus, the described system tends to reduce noise at signal frequency fIF. The signal transfer function Hx(ωX) exhibits near-unity gain at frequency fIF, so the signal is not significantly attenuated, thereby improving the signal-to-noise ratio (SNR) at the system output. Additionally, since fIF can be moved at will by adjustment of the phase shift in the regenerative or resonant loop 50, the band of high SNR can be readily adjusted.
FIG. 4 illustrates the response or performance of the arrangement of FIG. 1 with both the regenerative loop and the main degenerative loop closed. In FIG. 4, the gain of the signal transfer function Hx(ω) is plotted as dash line 412, and the noise transfer function He(ω) is 410. As illustrated, plot 412 is flat, and has essentially unity signal gain for all frequencies of interest, while plot 410 has a noise null at and near frequency fIF.
Equations (3) and (4) represent the general (regardless of order) transfer functions Hx(ω) and He(ω).
                                                                                          H                  xp                                ⁡                                  (                  s                  )                                            =                                                -                                                            H                      xp                                        ⁡                                          (                      s                      )                                                                      =                                                                            [                                                                                                    R                            nm                                                    ⁡                                                      (                            s                            )                                                                          ⁢                                                  e                                                      -                                                          sT                              e                                                                                                                          ]                                        p                                                        1                    +                                                                  [                                                                                                            H                              r                                                        ⁡                                                          (                              s                              )                                                                                ⁢                                                                                    R                              nm                                                        ⁡                                                          (                              s                              )                                                                                ⁢                                                      e                                                          -                                                              sT                                e                                                                                                                                    ]                                            p                                                                                                                                              =                                                                                          N                      p                                        ⁡                                          (                      s                      )                                                                                                  K                      p                                        ⁡                                          (                      s                      )                                                                      =                                  signal                  ⁢                                                                          ⁢                  transfer                  ⁢                                                                          ⁢                  function                                                                                        (        3        )            where:                Hx(s)=the signal transfer function of the continuous-time closed loop system;        Rnm(a)=continuous-time resonator transfer function for a resonator of order n, having m stages of cascading=Nnm(s)/Dnm(s);        and the additional subscript p indicates sampling rather than continuous time.        
                                                                                          H                  ep                                ⁡                                  (                  s                  )                                            =                                                1                                      1                    +                                          [                                                                                                    H                                                          r                              ⁢                                                                                                                                                                            ⁡                                                      (                            s                            )                                                                          ⁢                                                                              R                            nm                                                    ⁡                                                      (                            s                            )                                                                          ⁢                                                  e                                                      -                                                          siT                              e                                                                                                                          ]                                                                      =                                                                            D                      p                                        ⁡                                          (                      s                      )                                                                                                  K                      p                                        ⁡                                          (                      s                      )                                                                                                                                              =                              Noise                ⁢                                                                  ⁢                Shaping                ⁢                                                                  ⁢                Transfer                ⁢                                                                  ⁢                Function                                                                        (        4        )                            where:        He(s)=the noise transfer function of the continuous-time closed-loop system; and subscript p denotes sampling.        
In equations (3) and (4), the function Hr(s) represents the combined open-loop response of filters 42 and 44 of FIG. 1, where filter 42 represents switching operations within the digital-to-analog converter. More specifically, this open-loop response is given byHr(s)=HZOH(s)Heq(s)  (5)                where        HZOH(s)=Laplace Transform of zero order hold DAC reconstruction filter; and        Heq(s)=Laplace Transform of Elliptical Filter Required to Equalize the DAC zero order hold to produce a flat channel up to the IF frequency band selected.        
Equations (3) and (4) incorporate an aliasing operator denoted by subscripted bracketed functions of the form [ ]p, where
                                          [                          F              ⁡                              (                s                )                                      ]                    p                =                              k            s                    ⁢                                    ∑                              k                =                                  -                  ∞                                            ∞                        ⁢                          F              ⁡                              (                                  s                  -                                                            jω                      s                                        ⁢                    k                                                  )                                                                        (        6        )            and ks is a gain-normalizing constant. Operator [F(s)]p produces a periodic spectrum in response to a nonperiodic spectral input function. The periodicity of the aliasing function is fs, the sampling frequency. The effect of the aliasing operator is to sum the shifted components of the input spectrum over Nyquist bands, where Nyquist bands are defined as the spectra (nfS−fN) to (nfS+fN]), where n is an integer. In equations (3) and (4) one can define an effective resonance transfer function asReffp(s)=[Hr(s)Rnm(s)e−siTe]p  (7)Plot 240 of FIG. 2f represents the amplitude component of the effective resonance transfer function. The application of the aliasing operator to the open-loop transfer function combines the primary components of the resonance transfer function and the aliased components above fN in FIG. 2e. This resonance transfer function is perfectly periodic with frequency. For the first-order system of FIG. 1, the closed-loop transfer function is given by
                                          H            e                    ⁡                      (            s            )                          =                                            1              -                                                                    H                    ϕ                                    ⁡                                      (                    s                    )                                                  ⁢                                  H                  B                                ⁢                1                ⁢                                  (                  s                  )                                ⁢                                  H                  B                                ⁢                2                ⁢                                  (                  s                  )                                ⁢                                  e                                      -                                          sTD                      1                                                                                                                                                                1                    -                                                                                            H                          ϕ                                                ⁡                                                  (                          s                          )                                                                    ⁢                                              H                        B                                            ⁢                      1                      ⁢                                              (                        s                        )                                            ⁢                                              H                        B                                            ⁢                      2                      ⁢                                              (                        s                        )                                            ⁢                                              e                                                  -                                                      sTD                            1                                                                                                                +                                                                                                                                                                  H                        eq                                            ⁡                                              (                        s                        )                                                              ⁢                                                                  H                        ZOH                                            ⁡                                              (                        s                        )                                                              ⁢                                                                  H                        ϕ                                            ⁡                                              (                        s                        )                                                              ⁢                                          e                                              -                                                  sTD                          2                                                                                                                                                  =                                    D              ⁡                              (                s                )                                                    K              ⁡                              (                s                )                                                                        (        8        )            where:                e−sTpara=Laplace Transform of the parasitic delay, Tpara, which is produced by the components and interconnects in the primary feedback loop (e.g., Amplifiers, summing nodes, splitting nodes, A/D and DAC).        
In equation (8), the numerator produces a null response at the desired resonance frequency fIF, as indicated by plot 410 of FIG. 4. In the numerator term, the combined open-loop response of the regenerative loop 50 of FIG. 1 is aligned so that at the frequency of interest fIF, the magnitude of the response is unity, and the phase is zero degrees or a multiple of 360°. The denominator term of equation (8) includes the same function as is found in the numerator, plus the response of the open-loop main path 48 of FIG. 1. To achieve stability when the main and regenerative loops are closed, the individual responses of the loops are matched well enough to cancel each other. Equation (8) incorporates a polynomial multiplied by an exponential, and does not necessarily have a discrete solution, as would an equation with discrete poles and zeroes. Equations with discrete poles and zeroes include functions which are polynomials or a weighted sum of exponentials, but not a product of polynomials and exponentials. Thus, stability is not determined or defined by the locations of discrete poles in equation (8), but instead must be defined by a more general stability criterion. Control theory indicates that a return function T(s) can be defined as the denominator of the closed-loop transfer function K(s), minus one, T(s)=K(s)−1. Stability can be determined for such a closed-loop system from a Nyquist diagram of T(s). A Nyquist diagram has the real and imaginary parts of the return function plotted along corresponding mutually orthogonal axes, with s equal to jω, and ω swept over all frequencies. Stability is guaranteed when the Nyquist plot so made does not encircle the −1 point on the real axis. Thus, to guarantee stability, the open-loop transfer functions as defined above must be matched well enough in equation (8) to insure this condition. Ideally, when the regenerative and main open loop responses are perfectly matched, the Nyquist diagram will be a point at the origin. Such a response cannot realistically be achieved, but it can be approximated sufficiently for stability. The arrangement according to U.S. Pat. No. 5,608,400 allows this matching to be achieved in the presence of aliasing, which in turn allows stable sigma delta modulators to be made in which the desired signal gain and noise rejection are achieved at frequencies independent of the sampling rate.
FIG. 5 illustrates a sigma-delta analog-to-digital converter 510, which includes a higher-order embodiment of the resonator than does the arrangement of FIG. 1. The higher-order resonance structure implements multiple selectable poles and zeroes in the form of a single-stage recursive transversal filter (RTF). The poles are selected to enhance the resonance gain peak at a selected frequency or over a band of frequencies. The zeroes are selected for stability in the closed-loop condition, thereby providing increased attenuation at the frequency or band of frequencies in the noise transfer function He(ω) Elements of FIG. 5 corresponding to those of FIG. 1 are designated by like reference numerals. In FIG. 5, coupling arrangement 526 extending between output port 20o of summing circuit 20 and input port 32i of ADC 32 is different from coupling arrangement 26 of FIG. 1. Coupling arrangement 526 includes an input port 526i coupled to the output port 20o of summing circuit 20, and its third port 530 is coupled directly to its input port 526i. Coupling arrangement 526 of FIG. 5 includes a gain element 5641, a phase shifter 5631, a summing circuit 5701, and a time delay element 66, cascaded in the stated order between input port 526i and output port 526o. As in the case of the arrangement of FIG. 1, delay element 66 is within the main loop, but is not within a resonant loop. The placement of phase shifter 5631 in FIG. 5, on the other hand, is within the main degenerative feedback loop 48, but not within the multiple resonant loops.
The arrangement of FIG. 5 includes multiple regenerative or resonant loops, a first of which is designated 5501, and a second of which is designated 5502. The first resonant loop 5501 includes delay element 58, LPF 52, null filter 54, tap point 5721, gain element 601, a phase shifter 5621, summing circuits 5201 and 20, signal path 24, and third port 530. As mentioned previously in conjunction with FIG. 1, independent phase control of the main loop and a regenerative loop, such as of regenerative loop 5501, may be accomplished by means of separate phase shifters in each of the degenerative and main loops. Thus, phase shifter 5621 of regenerative loop 5501 is matched by phase shifter 5631 in a main loop portion 481; thus, degenerative main loop 48 in the arrangement of FIG. 5 includes a plurality of portions, all of which include ADC 32, DAC 38, DC blocking element 46, equalizer 44, path 16, summing circuits 12 and 20, and path 24 to port 530. All of the branches of main or degenerative loop 48 extend from port 30 of coupling circuit 526 to input port 32i of ADC 32. The first branch 481 of the main or degenerative loop 48 includes gain element 5641, phase shifter 5631, summing circuit 5701, and trim delay line 66. The second resonant loop 5502 includes delay element 58, LPF 52, null filter 54, tap point 5721, delay element 5581, tap point 5722, gain element 602, a phase shifter 5622, summing circuits 5202, 5201 and 20, signal path 24, and port 530. A corresponding second branch for phase control for the main loop is provided by a path extending from tap port 530 of coupler 526 to input port 32i of ADC 32, which path includes delay line 58 (if provided), LPF 52, null filter 54, tap point 5721, a gain control element 5642, and a phase shifter 5632 cascaded in a path extending between tap point 5721 and an input port of summing circuit 5702, and also includes summing circuits 5702 and 5701. As suggested by dash lines 5741 and 5742 extending from tap point 5722 and summing circuit 5202, respectively, of FIG. 5, additional resonant loops may be added by extending the recurrent structure of the illustrated resonant loops to further branches. As suggested by dash lines 5781 and 5782 extending from tap point 5722 and summing circuit 5702, respectively, additional corresponding branches of the main loop may be added in the same manner.
In equations (3) and (4), the effective resonance transfer functions are denoted by subscripts n and m, where n represents the order of the recursive transversal filter (resonator) of FIG. 5, and m represents the number of cascade stages in the RTF. Thus, m is unity in the case of FIG. 5. For orders (n) greater than one, the effective resonance is given by
                                          R                          n              ⁢                                                          ⁢              m                                ⁡                      (            s            )                          =                                                            ∏                                  i                  =                  1                                m                            ⁢                                                          ⁢                                                N                  i                                ⁡                                  (                  s                  )                                                                                    ∏                                  i                  =                  1                                m                            ⁢                                                D                  i                                ⁡                                  (                  s                  )                                                              =                                                    N                ⁡                                  (                  s                  )                                                            D                ⁡                                  (                  s                  )                                                      =                                          N                ⁡                                  (                  s                  )                                                            1                -                                  B                  ⁡                                      (                    s                    )                                                                                                          (        9        )            where
                              B          ⁡                      (            s            )                          =                                            ∏                              i                =                1                            m                        ⁢                          (                              1                -                                                      B                    i                                    ⁡                                      (                    s                    )                                                              )                                -          1                                    (        10        )                                                      B            i                    ⁡                      (            s            )                          =                                            H              Bi                        ⁡                          (              s              )                                ⁢                                    H              oi                        ⁡                          (              s              )                                ⁢                                    ∑                              l                =                1                                            L                i                                      ⁢                                          B                li                            ⁢                                                H                  Bli                                ⁡                                  (                  s                  )                                            ⁢                              e                                  -                                      slT                    e                                                                                                          (        11        )                                                      N            i                    ⁡                      (            s            )                          =                                            H              oi                        ⁡                          (              s              )                                ⁢                                    ∑                              l                =                1                                                              L                  i                                -                1                                      ⁢                                          A                li                            ⁢                                                H                                      A                    li                                                  ⁡                                  (                  s                  )                                            ⁢                              e                                                      -                                          s                      ⁡                                              (                                                  l                          -                          r                                                )                                                                              ⁢                                      T                    e                                                                                                          (        12        )            This higher order of the effective resonance will produce increased noise attenuation over a specified bandwidth centered at a desired frequency fIF, as shown by plot 642 of FIG. 6c. Stability is insured in the multipole context by a criterion similar to that used in the first-order system, namely that of equation (8). In the multipole context of FIG. 5, the denominator of the closed-loop transfer function is given byK(s)=1−B(s)+N(s)e−slTe  (13)so that the return function is given byT(s)=N(s)e−sTe−B(s)  (14)To illustrate the above-mentioned stability criterion in relation to equation (12), a selection for the weights used in paths 5501, 5502, . . . and the weights used in corresponding feedforward paths 481, 482 of recursive transversal filter or resonator 50 of FIG. 5 might be such that the weight of each feedforward path (481) is equal to the weight of the corresponding feedback path (5501) When this equal-weight condition prevails, the idealized closed-loop noise transfer function exhibits a finite-impulse-response (FIR) characteristic such that a Nyquist diagram of the response is represented as a point at the origin. Mismatches are to be expected between the functions N(s) and B(s), arising from the actual, as opposed to theoretical, performance of the bandlimiting filters 42, 44, 52, and 54 of FIG. 5, and from gain and phase variation of components (amplifiers, attenuators, delay lines, etc. which are used. Note that it is not possible to ascribe the function N(s) solely to filters 42 and 44, because filters 52 and 54 affect N(s). Thus, in practice the stability of the arrangement of FIG. 5 is guaranteed by requiring adequate matching of the filter characteristics, to avoid encirclement of the point −1 on the real axis of the Nyquist diagram when the weighting is applied. Once the bandlimiting filters have been designed for stability using FIR weighting in the RTF as described above, the FIR filter weighting can be replaced by IIR weighting, and stability is maintained.
FIG. 6a illustrates an amplitude-frequency spectrum of an analog input signal 610 centered at a frequency fIF, which may be applied to the sigma-delta analog-to-digital converter 510 of FIG. 5. FIG. 6b illustrates the open-loop spectral response 612 of the resonant loops 5501, 5502, . . . of the arrangement of FIG. 5, as affected by combining the regenerative loop responses with the response of low-pass filter 52 and the response of null-filter 54. As illustrated, response 612 includes a multiple-pole gain peak 614, resulting from the multiple resonant poles, extending over a bandwidth about fIF which includes the analog frequencies. This open-loop gain peak, when both the regenerative and degenerative loops are closed, results in low through loss at the signal frequencies. Plot 612 also includes additional poles 616 and 618, in which pole 616 is below the Nyquist frequency fN, and pole 618, which is above the Nyquist frequency, has a gain of less than the maximum stable gain, which is represented by line 630. FIG. 6c illustrates by a plot 640 the gain of the analog-to-digital converter 510 of FIG. 5, with all the resonant and degenerative feedback loops closed. The solid-line plot 642 of FIG. 6c represents the response of the loop on noise e(n) injected into port 32n of ADC 32 of FIG. 5, with a multipole null centered on frequency fIF, to suppress the noise at system output path 40. The dash-line path 644 illustrates the overall gain response of the sigma-delta analog-to-digital converter 510 to the signal. As illustrated, the signal gain is much greater than the gain applied to the noise. Ideally, the signal gain is unity. As described above in relation to FIG. 1, the multiple resonant frequencies at which the noise rejection is high in null 642 of FIG. 6c can be adjusted by selecting the time delays of delay elements 58, 66, 5581, . . . , and, in pairs, the phase shifts of phase shifters 5621, 5631; 5622, 5632; . . .
In the ordinary sigma-delta ADC, the two bandwidth-draping functions are implemented as recursive transversal filters. These RTF provide a tradeoff between depth of filtering and latency or time delay, because more elements are needed in the filter for greater depth of filtering, but the additional elements add to the delay.
As it so happens, in present-day sigma-delta analog-to-digital converters, the latency of the loop filter or resonator is on the same order of magnitude as the latency of the combination of the ADC and the DAC in the feedback loop, and as a result, both contribute to the overall loop delay, and together limit the maximum bandwidth.
Wideband sigma-delta analog-to-digital converters, andor improved or alternative continuous-time loop filter sigma-delta modulators that enable wider pass band bandwidths are desired.